Computing bounded depth decompositions is a bottleneck in many applications of the treedepth parameter. The fastest known algorithm, which is due to Reidl, Rossmanith, S\'{a}nchez Villaamil, and Sikdar [ICALP 2014], runs in $2^{\mathcal{O}(k^2)}\cdot n$ time and it is a big open problem whether the dependency on $k$ can be improved to $2^{o(k^2)}\cdot n^{\mathcal{O}(1)}$. We show that the related problem of finding DFS trees of bounded height can be solved faster in $2^{\mathcal{O}(k \log k)}\cdot n$ time. As DFS trees are treedepth decompositions, this circumvents the above mentioned bottleneck for this subclass of graphs of bounded treedepth. This problem has recently found attention independently under the name Minimum Height Lineal Topology (MinHLT) and our algorithm gives a positive answer to an open problem posed by Golovach [Dagstuhl Reports, 2023]. We complement our main result by studying the complexity of MinHLT and related problems in several other settings. First, we show that it remains NP-complete on chordal graphs, and give an FPT-algorithm on chordal graphs for the dual problem, asking for a DFS tree of height at most $n-k$, parameterized by $k$. The parameterized complexity of Dual MinHLT on general graphs is wide open. Lastly, we show that Dual MinHLT and two other problems concerned with finding DFS trees with few or many leaves are FPT parameterized by $k$ plus the treewidth of the input graph.

翻译：计算有界深度分解是树深参数在许多应用中的瓶颈。已知最快的算法由Reidl、Rossmanith、Sánchez Villaamil和Sikdar [ICALP 2014]提出，运行时间为$2^{\mathcal{O}(k^2)}\cdot n$，而能否将$k$的依赖改进为$2^{o(k^2)}\cdot n^{\mathcal{O}(1)}$是一个重要的开放问题。我们证明，寻找有界高度DFS树的相关问题可以在$2^{\mathcal{O}(k \log k)}\cdot n$时间内更快地解决。由于DFS树即是树深分解，这为有界树深图的这一子类绕过了上述瓶颈。该问题最近以最小高度线形拓扑（MinHLT）的名称独立受到关注，我们的算法对Golovach [Dagstuhl Reports, 2023]提出的开放问题给出了肯定回答。我们通过研究MinHLT及相关问题在其他几种设置下的复杂性来补充主要结果。首先，我们证明该问题在弦图上保持NP完全性，并针对对偶问题（要求DFS树高度至多为$n-k$，以$k$为参数）在弦图上给出一个FPT算法。对偶MinHLT在一般图上的参数化复杂性仍是广泛开放的。最后，我们证明对偶MinHLT以及另外两个涉及寻找叶子节点少或多DFS树的问题，在参数$k$加上输入图的树宽时是FPT的。